## Duke Mathematical Journal

### Serrin’s overdetermined problem and constant mean curvature surfaces

#### Abstract

For all $N\geq9$, we find smooth entire epigraphs in $\mathbb{R}^{N}$, namely, smooth domains of the form $\Omega:=\{x\in\mathbb{R}^{N}|x_{N}\gt F(x_{1},\ldots,x_{N-1})\}$, which are not half-spaces and in which a problem of the form $\Delta u+f(u)=0$ in $\Omega$ has a positive, bounded solution with $0$ Dirichlet boundary data and constant Neumann boundary data on $\partial\Omega$. This answers negatively for large dimensions a question by Berestycki, Caffarelli, and Nirenberg. In 1971, Serrin proved that a bounded domain where such an overdetermined problem is solvable must be a ball, in analogy to a famous result by Alexandrov that states that an embedded compact surface with constant mean curvature (CMC) in Euclidean space must be a sphere. In lower dimensions we succeed in providing examples for domains whose boundary is close to large dilations of a given CMC surface where Serrin’s overdetermined problem is solvable.

#### Article information

Source
Duke Math. J., Volume 164, Number 14 (2015), 2643-2722.

Dates
Received: 15 October 2013
Revised: 9 November 2014
First available in Project Euclid: 26 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1445865570

Digital Object Identifier
doi:10.1215/00127094-3146710

Mathematical Reviews number (MathSciNet)
MR3417183

Zentralblatt MATH identifier
1342.35188

#### Citation

Del Pino, Manuel; Pacard, Frank; Wei, Juncheng. Serrin’s overdetermined problem and constant mean curvature surfaces. Duke Math. J. 164 (2015), no. 14, 2643--2722. doi:10.1215/00127094-3146710. https://projecteuclid.org/euclid.dmj/1445865570

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